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Products, quotients and roots of complex numbers in polar form. De Moivre’s theorem. Roots of unity

Polar representation of a complex number. The polar representation of the complex number

z = x + iy

is

z = x + iy = r (cos θ + i sin θ)

where

x = r cos θ

y = r sin θ

θ = arctan (y/x)

See Figure 1. The radius vector r is called the modulus or absolute value of the complex number and the polar angle θ is called the amplitude or argument of the number. The argument θ of a complex number z is often denoted by arg z. The abbreviation cis θ is sometimes used for cos θ + i sin θ. In polar coordinates a point P is often specified by the number pair (r, θ).

Product of two complex numbers in polar form. The product z_{1}z_{2} of the two
complex numbers

z_{1} = r_{1} (cos θ_{1} + i sin θ_{1})

and

z_{2} = r_{2} (cos θ_{2} + i sin θ_{2})

is

1) z_{1}z_{2} = r_{1}r_{2}[cos (θ_{1} + θ_{2}) + i sin (θ_{1} + θ_{2})] ,

a result which can be easily obtained by utilizing the trigonometric identities

sin(A B) = sin A cos B cos A sin B

cos(A B) = cos A cos B sin A sin B .

Quotient of two complex numbers in polar form. The quotient z_{1}/z_{2} of the
two complex numbers

z_{1} = r_{1} (cos θ_{1} + i sin θ_{1})

and

z_{2} = r_{2} (cos θ_{2} + i sin θ_{2})

is

2) z_{1}/z_{2} = ( r_{1}/r_{2})_{ }[cos (θ_{1} - θ_{2}) + i sin (θ_{1} - θ_{2})]

De Moivre’s Theorem. For any complex number z = r( cos θ + i sin θ )

3) z^{n} = [ r(cos θ + i sin θ)]^{n} = r^{n} (cos nθ + i sin n)

This formula holds for every real value of the exponent n. For example, if the exponent is a fraction 1/n, we get

Rules of arguments. If

z_{1} = r_{1} (cos θ_{1} + i sin θ_{1})

z_{2} = r_{2} (cos θ_{2} + i sin θ_{2})

w = z^{a}

where a is a real number, then by 1), 2) and 3 ) above

1] arg (z_{1}z_{2}) = arg z_{1} + arg z_{2}

2] arg (z_{1}/z_{2}) = arg z_{1} - arg z_{2}

3] arg w = a arg z

Def. N-th root of a
number. Let n be a
positive integer. If a^{n} = b,
then a is said to be the n-th
root of b.

Roots of complex numbers in polar form. The n distinct n-th roots of the complex number

z = r( cos θ + i sin θ)

can be found by substituting successively k = 0, 1, 2, ... , (n-1) in the formula

The n roots are equally spaced around the circumference of a circle in the complex plane. See Figure 2. If the complex number for which we are computing the n n-th roots is z = ρ( cos θ + i sin θ) the radius of the circle will be

and the first root w_{0} corresponding to k = 0 will be at an amplitude of α = θ/n. This root will be
followed by the n-1 remaining roots at equal distances apart. The angular amplitude between
each root is Δα = 360^{o}/n.

Example. Suppose we wish
to compute the ten 10-th roots
of z = 8(cos 150^{o} + i sin 150^{o})
shown in Fig. 3. The ten
roots w_{0}, w_{1}, .... , w_{9} would be
spaced evenly around a circle
as shown in Fig. 4. The first
root, w_{0}, would be at an
amplitude of α = 150/10 = 15^{o}.
The rest of the roots would be
spaced at Δα = 360/10 = 36^{o}
intervals.

Roots of unity. The n n-th roots of 1 are obtained from 5) above by letting r = 1 and θ = 0. They are

for k = 0, 1, 2, ... , (n-1)

Let us denote the root corresponding to k = 1 by w. This root w is then given by

The n n-th roots of 1 then correspond to powers of w:

w, w^{2}, w^{3}, ... ,w^{n}

where w^{n} = 1

The roots are equally spaced around the circumference of a unit circle in the complex plane. See Figure 5.

Primitive roots of unity. Of the n n-th roots of 1 some of the roots may be m-th roots of 1 where m is some integer less than n. For example, the 6 sixth roots of 1 are

r_{1} = w

r_{2} = w^{2}

r_{3} = w^{3}

r_{4} = w^{4}

r_{5} = w^{5}

r_{6} = w^{6} = 1

Of these, r_{3} = w^{3} and r_{6} = w^{6} are square roots of 1 and r_{2} = w^{2}, r_{4} = w^{4} and r_{6} = w^{6} are
cube roots of 1. The primitive roots of 1 are those roots which are not m-th roots of 1 for some 0
< m < n. Thus in the example just given the roots r_{1} = w and r_{5} = w^{5} are primitive roots of
1. In other words, of the n n-th roots of 1, a particular n-th root r is a primitive root if and only if
r^{m}
1 for any integer m less than n.

Theorem Let w, w^{2}, w^{3}, ... ,w^{n} be the n n-th roots of 1. Let m be any integer 0 < m < n and
let d be the greatest common divisor (m,n) of m and n. If d > 1 then , w^{m} is an n/d-th root of 1.

Example. Let m = 3 and n = 6. Then d = (m,n) = (3,6) = 3 and n/d = 2. Thus w^{3} is a square
root of 1.

Corollary. The primitive n-th roots of 1 are those and only those n-th roots w, w^{2}, w^{3}, ... ,w^{n} of
1 whose exponents are relatively prime to n.

Roots of a complex number in terms of the roots of unity. Let

z = a + bi = r( cos θ + i sin θ )

and

Then the n n-th roots of z are

z_{0}, wz_{0}, w^{2}z_{0}, ... ,w ^{k-1}z_{0}

where

References.

James & James. Mathematics Dictionary.

Brink. A First Year of College Mathematics.

Spiegel. College Algebra.

Hauser. Complex Variables with Physical Applications.

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